In 1980 the Curie brothers observed that when a permanently polarized material is subjected to a mechanical force an electric field is produced. Today, this phenomenon is referred to as the piezoelectric effect. Conversely, when an alternating electric current is applied to opposite faces of a piezoelectric crystal, the crystal expands and contracts in concert with the alternating electric current. Piezoelectric crystals typically resonate within narrowly defined frequency ranges and when suitably mounted they can be used in electric circuits as components of highly selective filters or as frequency-control devices for very stable oscillators.
As demand for space on the available radio-frequency spectrum grows, spacing between assigned frequencies has become tighter. To avoid interference between transmitters operating in the same geographical area or on adjacent channels, it is essential that frequency be accurately controlled.
The internal constraints of communication devices also demand small frequency deviation as a function of temperature, and as consumer demand continually drives down the size and the cost of this equipment, the need for smaller piezoelectric devices that can maintain tight temperature performance and that are less costly to manufacture becomes increasingly greater.
Due to its high Q value (a measure of the properties that give the quartz crystal its unique ability to control a frequency) and its low cost, quartz (SiO2) is the material of choice for the production of piezoelectric devices. Quartz plates are commonly used for frequency control in electronic devices such as computers, cellular phones, pagers, radios, and wireless data devices, and have been exploited to produce very accurate clocks.
Originally, quartz plates were made from natural quartz crystals, but today quartz plates are cut almost exclusively from synthetically produced quartz crystals. The orientation, or angle, of the cut with respect to the crystallographic axes X, Y, and Z (FIG. 1) determines how the oscillation frequency of the plate will be affected by temperature variations where temperature dependence is often expressed in terms of the temperature coefficient of frequency and by other properties of the resonator. In addition, the angle of cut determines the tolerance level, or margin of error, for a given cut. In mass production a low margin of error results in substantial increases in manufacturing costs and in the additional cost of providing corrections for errors made.
The X, Y, and Z crystallographic axis are often referred to as the electrical, mechanical, and optic axes, respectively. The earliest quartz crystal cut was made with the crystal's major face perpendicular to the X-axis and with its length along the Y-axis. Whenever a cut is made so that the major face of the crystal plate is perpendicular to a particular axis, that cut is given the name of the axis to which it is perpendicular (FIG. 2). For example, when a cut in made so that the crystal's major face is perpendicular to the X-axis, it is referred to as an X cut.
To eliminate the coupling effect that is inherent in X cuts, the angle of the X cut can be rotated 18.5° from the Y-axis toward the Z axis (FIG. 3). The rotated X cut gives a good resonance free from other modes. It suffers, though, from the fact that, to get a reasonably high frequency from this type of length vibrating cut, too small a length to be practical is required.
Eventually a Y cut (a cut where the large crystal face is perpendicular to the Y axis) was developed. This cut, however, suffers from large changes in frequency with changes in temperature, i.e., the frequency increases about 86 ppm (parts per million) for each degree Centigrade increase in temperature. To reduce the temperature effect, the angles at which the Y cuts are made can be varied. When cuts are made at angles of either +35° 15′ or −49° (rotated about the X axis) the first order, or linear portion, of the temperature coefficient of frequency is zero. A quartz plate cut at a +35° 15′ angle is referred to as an AT cut plate and a plate cut at a −49° angle is referred to as a BT cut plate (FIG. 4).
The conventional mathematical description of quartz's piezoelectric behavior as a function of temperature dates back to the early WWII efforts and is based on several simplifying mathematical assumptions. These simplifying assumptions (i.e., ignoring many of the properties that control the piezoelectric behavior of quartz) enable easy, though not definitive, calculation of the temperature coefficients of various cuts of quartz plates. For example, the original mathematical description published by Heising in 1946 (Quartz Crystals for Electrical Circuits, Raymond A. Heising, D. Van Nostrand Co. Inc. 1946, pg. 27), and republished in a restated form later by Salt (Hy-Q Handbook of Quartz, David A. Salt, 1983), is based on the assumption that all piezoids of quartz are equivalent and that their stiffness coefficients repeat 0° to 90°, 90° to 180°, 180° to 270°, and 270° to 360°. It is also based on the assumption that the mechanical vibrations in the crystal are ideal and perfectly elastic.
These assumptions led to the published (Quartz Crystals for Electrical Circuits, Raymond A. Heising, D. Van Nostrand Co. Inc. 1946 pg. 54) and commonly accepted equation:
                                                                        T                f                            =                            ⁢                              3.9                +                                  6.5                  ⁢                                                                          ⁢                                      cos                    2                                    ⁢                  θ                                +                                                                                                      ⁢                                                1                  2                                ⁡                                  [                                                                                                              c                          66                                                ⁢                                                  T                                                      c                            66                                                                          ⁢                                                  sin                          2                                                ⁢                        θ                                            +                                                                        c                          44                                                ⁢                                                  T                                                      c                            44                                                                          ⁢                                                  cos                          2                                                ⁢                        θ                                            +                                                                        T                                                      c                            14                                                                          ⁢                                                  c                          14                                                ⁢                        sin                        ⁢                                                                                                  ⁢                        2                        ⁢                                                                                                  ⁢                        θ                                                                                                                                      c                          66                                                ⁢                                                  sin                          2                                                ⁢                        θ                                            +                                                                        c                          44                                                ⁢                                                  cos                          2                                                ⁢                        θ                                            +                                                                        c                          14                                                ⁢                        sin                        ⁢                                                                                                  ⁢                        2                        ⁢                                                                                                  ⁢                        θ                                                                              ]                                                                                        (        1        )            where:                Tƒ=frequency temperature coefficient,        θ=angle of rotation from the Z axis,        cxx=is the value of stiffness. The subscripts denote the stiffness of a given rhombohedral axis.        
This equation uses the older IRE (Institute of Radio Engineers) convention of designating the “AT” cut as a positively rotated Y cut. Subsequent publications by Bottom (Introduction to Quartz Crystal Design, Virgil Bottom, Van Nostrand Reinhold Co. 1982) and by Salt (Hy-Q Handbook of Quartz, David A. Salt, 1983) updated the formulation to use the more modern sign convention that produces the familiar curve giving the relationship of temperature coefficient to angle of cut (FIG. 5). This graph shows zero frequency temperature coefficient points for the first order approximation at the familiar AT and BT cut angles. The commercial success of the AT cut in comparison to the BT cut lies in the fact that for the AT cut not only is the first order temperature coefficient zero but its second order temperature coefficient is also zero. This condition gives the AT cut a much lower total frequency deviation as a function of temperature, compared to the BT cut.
The AT cut provides a low cost quartz plate with good frequency-temperature performance. The AT cut, although widely used, does not perform well when mechanically and thermally stressed. This shortcoming can be reduced by using a doubly rotated cut. A doubly rotated cut is obtained by starting with a traditionally rotated Y cut, with its initial rotation about the X axis through an angle θ, followed by a second rotation about the new Z axis (referred to as Z′ and is defined by the crystallographic axes of the rotated plate) through an angle φ. Thus, a doubly rotated cut is defined by the angles φ and θ, where in the case of an AT cut the φ is zero.
A third rotation, referred to as an in-plane omega rotation, is sometimes used to separate unwanted vibrational modes from the main mode. An omega rotation is made in the plane of either a singly or a doubly rotated cut. For example, after a plate is oriented by a rotation oft theta equal to 44 degrees and another with phi equal to 12 degrees, the plate is then rotated within the plane so that the length of the plate is not exactly along either the x or z axis, but resides somewhere between the x and z axes.
These improvements none withstanding, the modern trend is for tighter and tighter frequency-temperature performance from piezoelectric resonators. This requirement has resulted in the use of external temperature compensation schemes being applied to systems utilizing AT-cut quartz resonators, which increases cost, time, and system complexity. There are other angles of cut, but they all suffer from either strong frequency dependence on temperature or on the presence of competing modes of vibrations.
It is clear then that there exits a need for additional angles of cut that will reduce the frequency dependence on temperature and on the presence of other modes of vibrations. It is also clear that there is a need for a way to determine, without undue experimentation, which angles of cut will produce desired results.